Spin caloritronic nano-oscillator

Energy loss due to ohmic heating is a major bottleneck limiting down-scaling and speed of nano-electronic devices, and harvesting ohmic heat for signal processing is a major challenge in modern electronics. Here, we demonstrate that thermal gradients arising from ohmic heating can be utilized for excitation of coherent auto-oscillations of magnetization and for generation of tunable microwave signals. The heat-driven dynamics is observed in Y3Fe5O12/Pt bilayer nanowires where ohmic heating of the Pt layer results in injection of pure spin current into the Y3Fe5O12 layer. This leads to excitation of auto-oscillations of the Y3Fe5O12 magnetization and generation of coherent microwave radiation. Our work paves the way towards spin caloritronic devices for microwave and magnonic applications.

The right ordinate axis shows the corresponding wire temperature that was obtained from measurements of the wire resistance as a function of bath temperature at small bias current.   Figure 4: Microwave emission measurements. a Schematic of the experimental setup for measurements of spin torque oscillator (STO) microwave emission with magnetic field modulation. The nanowire with magnetic field H applied at an angle ϕ is supplied with a direct current I dc via a bias tee. The microwave signal is amplified and detected using a spectrum analyzer. Its video-out signal is processed with a lock-in amplifier. The reference-out signal is used to generate a modulation field Hac using an audio amplifier. b-e Microwave emission spectra from a 90 nm wide YIG(23 nm)/Pt(8 nm) nanowire STO with a 0.9 µm long active region measured at H near 1.5 kOe, ϕ = 70 • and I dc = 1 mA by different techniques. b Conventional technique without field modulation. c Field modulation technique at fixed magnetic field and swept frequency. In order to estimate the magnitude of thermal gradients arising from ohmic heating, we carried out finite element simulations of coupled electrical and thermal transport in the YIG/Pt nanowire devices using COMSOL Multiphysics package [2]. We employed a fully realistic three-dimensional nanowire device geometry illustrated in Supplementary In these simulations, we used the temperature-dependent heat conductivities and heat capacities of YIG and GGG as reported in Ref. 3. The Pt layer resistivity in our YIG(23 nm)/Pt(8 nm) nanowire devices was measured in the temperature range from 140 K to 300 K and found to be linear as expected: ρ(T ) = ρ 0 (1 + αT ) with ρ 0 = 3.35 × 10 −7 Ω·m and α = 1.59 × 10 −3 K −1 , which is similar to previously reported values in thin Pt films [4,5]. The temperature-dependent heat capacity of Pt reported in Ref. 6 was employed in the simulations, and the thermal conductivity of the Pt layer was calculated from its electrical conductivity via the Wiedemann-Franz law. Literature values of the thermal and electrical conductivity and heat capacity of the lead materials were employed [2].
Supplementary Figure 2a shows the calculated spatial distribution of temperature in the YIG/Pt nanowire device studied in this work at the bath temperature of 140 K and direct current bias I dc = 2.5 mA that is similar to the critical current. Supplementary Figures 2b and 2c show the depth profiles of the temperature in the center of the YIG/Pt wire. These figures reveal that the Pt layer temperature rises to 220 K. The temperature in the YIG layer rapidly decreases with depth resulting in a large temperature gradient ∇T = 0.26 K nm −1 across the YIG layer thickness. This high degree of ohmic heating and the large value of ∇T result from the high resistivity of the Pt layer and efficient heat channeling into the YIG underlayer in the nanowire geometry employed in our experiment. In this geometry, the metallic leads do not function as efficient heat sinks because their overlap area with the nanowire is relatively small, which results in a high degree of ohmic heating of the Pt nanowire and dissipation of this heat is mainly through the GGG/YIG underlayers. The quasi-one-dimensional nature of the Pt nanowire heat source and the three-dimensional character of the heat flow in the GGG substrate further enhance ∇T across the thickness of the YIG layer.
The validity of these COMSOL simulations can be directly checked against the experiment because the temperature of the Pt wire can be determined by measuring its resistance. Supplementary Figure 3 shows the resistance of the Pt nanowire measured as a function of direct current bias. These data and the linear relation between the Pt nanowire resistance and temperature reveal that the Pt nanowire temperature at the bath temperature of 140 K and I dc = 2.5 mA is 260 K. We therefore conclude that the COMSOL simulations underestimate the degree of ohmic heating of the Pt wire and that the actual temperature gradient across the YIG film thickness is likely to exceed that predicted by the simulations.
We also employed COMSOL simulations to evaluate the ohmic heating in the YIG/Pt microdisk spin torque oscillators investigated in Ref. defined on top of a GGG substrate. Two Ti(20 nm)/Au(80 nm) leads are attached to the disk with the inter-lead gap of 1 µm. The system is covered with an SiO 2 (300 nm)/Au(250 nm) bilayer (not shown in Supplementary Figure 2d). The Pt layer resistivity ρ = 1.7 × 10 −7 Ω·m was directly measured for this system [1], and thermal conductivity of Pt was calculated via the Wiedemann-Franz law. In these simulations we also use the experimental parameters of Ref. 1: a bath temperature of 293 K and a critical current of 7.4 mA. Literature values of the temperature-dependent electrical conductivity, thermal conductivity and heat capacity of Au, Ti, SiO 2 , YIG and GGG [2,3,6] were employed in the simulations.
Supplementary Figures 2e,f show the depth dependence of the temperature in the center of the disk at I dc = 7.4 mA. It is clear from these figures that ohmic heating of the Pt layer is substantially smaller than in our nanowire devices due to the lower electrical resistivity of the Pt layer employed in Ref. 1 and better heat sinking by the Ti/Au leads having significant contact area with the microdisk. Combined with this, these devices are encased by 300 nm of silicon oxide to allow electrical isolation from a 250 nm thick Au antenna, providing further heat sinking. The resulting temperature gradient in the YIG film across its thickness at the critical current is only 0.033 K nm −1 -an order of magnitude smaller than that in our nanowire devices. Therefore, it is not surprising that the antidamping torque in these devices predominantly arises from spin Hall current with a negligible contribution from spin Seebeck current as evidenced by the 1/ sin ϕ angular dependence of the critical current observed for these samples [1].

Supplementary note 2. Field modulated detection of microwave emission.
Supplementary Figure 4a schematically illustrates the experimental setup employed in our field-modulated microwave emission measurements. A low-frequency (∼1 kHz), small-amplitude harmonic magnetic field H ac is applied to an STO sample parallel to a constant external magnetic field H dc . A direct current bias I dc is supplied from a custom built low noise current source and applied to the STO via a Picosecond 5541A-104 bias tee. The direct bias current excites the self-oscillations of magnetization. The microwave signal emitted by the sample is then amplified through a Miteq AMF-6F-00100400-10-10P low noise microwave amplifier with 62 dB gain, noise figure of 1 dB, and frequency band of 0.1-4 GHz. The amplified signal is then sent to an Agilent E4408B spectrum analyzer, configured in a single-frequency continuous detection mode. This configuration measures the integrated microwave power in a 5 MHz bandwidth around a fixed measurement frequency. The STO generation frequency is modulated by H ac , which results in a modulation of the STO power at the measurement frequency. The modulated STO emission power dP dH is measured by a Signal Recovery 7225 lock-in amplifier via a video output port of the spectrum analyzer. In order to obtain a field-modulated STO emission spectrum, the data is collected point-by-point by stepping the measurement frequency of the spectrum analyzer over a desired frequency range.
The conventional method of measuring STO microwave emission spectrum, in which the emission power is simply recorded as a function of frequency, did not yield a signal exceeding the noise floor for the 350 nm wide YIG/Pt nanowire samples discussed, and our field modulation technique was required to observe the signal. In order to quantitatively compare the field-modulated emission method to the conventional method, we employ an STO sample based on a 90 nm wide YIG(23 nm)/Pt(8 nm) nanowire with a 0.9 µm long active region. This STO generates higher microwave signals, which can be measured by the conventional technique as illustrated in Supplementary Figure 4b. Supplementary Figures 4b,c directly compare the microwave emission spectra for conventional and field-modulated detection measured under identical conditions (H = 1.5 kOe, ϕ = 70 • , I dc = 1 mA, measurement time 17 minutes). The conventional method gives a spectral peak with integrated power of 21 fW. In contrast, the field modulation method yields a prominent dP dH signal with high signal-to-noise ratio and the line shape similar to a Lorentzian curve derivative as illustrated in Supplementary Figure 4c. Supplementary Figures 4d,e illustrate that the field modulation method can be further improved by sweeping external magnetic field instead of stepping the center-frequency as done in Supplementary Figure 4c. In Supplementary  Figure 4d, the field-modulated emission signal is measured as a function of applied field giving the expected antisymmetric line shape. This signal can be directly integrated in magnetic field yielding a symmetric emission curve as a function of magnetic field. The development of this microwave detection technique increases the signal-to-noise ratio by two orders of magnitude, allowing detection of ultra low-level microwave signals emitted by magnetic devices.
Approximate calibration of the power scale for the field-integrated spectra such as that shown in Supplementary  Figure 4e can be performed via comparison of the spectral peak amplitudes in Supplementary Figures 4b,e. We estimate the maximum power spectral density P max generated by the 350 nm wide YIG/Pt nanowire shown in Fig. 2b to be approximately 0.1 fW MHz −1 and the corresponding integrated power to be approximately 6 fW.
We also note that the nanowire geometry can be used to tune both the frequency and the amplitude of the microwave signal generated by the YIG/Pt nanowire STO. We find that decreasing the width of the nanowire from 350 nm to 90 nm results in a decrease of the generated signal frequency. At the same time, the output power of the STO increases by over a factor of three. The decrease of the resonance frequency results from a higher demagnetization field in the narrower nanowire [7] while the increase of the output power can be attributed to a larger volume fraction of the wire occupied by the spin wave mode (see Fig. 4c).

Supplementary Note 3. Auto-oscillation amplitude.
The precession cone angle of the auto-oscillatory YIG magnetization can be estimated from the output microwave power of the YIG/Pt nanowire STO. The integrated microwave power P int generated by an STO is proportional to the square of the direct current bias I dc and the amplitude of resistance auto-oscillations δR ac [8,9]: where R is the sample resistance and R 50 is the 50 Ω microwave transmission line impedance. Assuming the angular dependence of the YIG/Pt nanowire resistance is R = R 0 + ∆R 2 cos 2ϕ as expected for SMR, the small-amplitude dynamic resistance oscillations δR ac are related to the in-plane precession cone angle ϕ c in the macrospin approximation as: where ϕ 0 is the equilibrium direction of the YIG magnetization. The maximum value of ϕ c achieved by the YIG magnetization in the 350 nm wide nanowire device can be calculated from Equations 1 and 2 by using the generated integrated power P int = 6 fW and ∆R = 0.05 Ω extracted from Fig. 1b. This calculation gives the precession cone angle in the macrospin approximation ϕ c ≈ 6 • . Taking into account that the excited LF mode has the edge character as shown in Fig. 4c, and that the edge mode occupies approximately one third of the nanowire volume as predicted by our micromagnetic simulations, the amplitude of the YIG magnetization oscillations at the nanowire edge is estimated to be approximately 20 • . We stress that this is merely an estimate because contributions to the generated microwave signal beyond SMR such as inductive signal generated by precessing magnetization [1] can be non-negligible in our nanowire devices.

Supplementary Note 4. Micromagnetic simulations
Micromagnetic simulations of the spin wave eigenmode frequencies of the YIG/Pt nanowires were performed using a modified version of the finite-differences simulation code MuMax 3 [10]. The nanowire was discretized into 2048×32×4 cells, resulting in a cell size of 6.30 × 8.75 × 7.50 nm 3 . The saturation magnetization M s = 130 kA m −1 [11] and the exchange constant A ex = 3.5 pJ m −1 were used [12]. The spin wave eigenfrequencies were determined as the peak position of the Fourier-transform of the dynamic magnetization excited by a sinc-shaped magnetic field pulse [13]. The simulation time was chosen to be 25 ns, which results in an FFT frequency resolution of 40 MHz. The spin wave profiles shown in Fig. 4c are represented by the cell-specific Fourier amplitude.

Supplementary Note 5. Magnetic damping.
We use the conventional broadband FMR technique [14] to measure the FMR linewidth ∆H (defined as half width at half maximum of the Lorentzian absorption curve) as a function of frequency f for YIG(23 nm) and YIG(23 nm)/Pt(8 nm) films. The damping constant α and the inhomogeneous broadening parameter ∆H 0 are determined from the slope and zero-frequency intercept of the ∆H(f ) data [15]. These measurements give α = 0.0014 and ∆H 0 = 1.1 Oe for the YIG film and α = 0.0035 and ∆H 0 = 2.6 Oe for the YIG/Pt bilayer. The linewidth ∆H measured at 3.2 GHz for the YIG/Pt bilayer film is found to be 6.7 Oe, which is similar to ∆H = 6 Oe at 3.2 GHz measured in the YIG/Pt nanowire device by ST-FMR at the lowest microwave power value (−5 dBm) as shown in Fig. 4e. This demonstrates that patterning of the YIG/Pt film into the nanowire device does not significantly alter the YIG layer damping.